1636
Topological Representation of Stereoisomerism. I. Polytopal Rearrangements E. L. Muetterties Contribution from the Frick Laboratory, Princeton University, Princeton, New Jersey 08540. Received August 14, 1968
Abstract: This paper is an attempt to establish principles and definitions that will have general applicability in the area of topological representation of stereoisomerism in molecular species. The present analysis is restricted to polytopal rearrangements. Illustrative examples are taken from inorganic and organometallic chemistry.
T
his is a first attempt’ to outline a general approach to topological representations of stereoisomerization processes. Definitions and illustrative examples are developed for a polytopal isomerization process which is defined below. The ultimate objectives are (1) delineation of differences in stereoisomerizations for alternative mechanisms, (2) simple pictorial representations that illustrate subgroups and depict relative energy relationships, and (3) clues to render molecular structures stereochemically more, or less, rigid.’”
Polytopal Isomers and Rearrangements Stereoisomers apply to different spatial arrangements of atoms in molecules or ions of the same constitution.’ A special class of stereoisomers, critical to the following discussion, is designated as polytopal isomers, G. This class differentiates spatial arrangements for any constitutional form in terms of idealized polygons or polyhedra whose vertices are defined3 by the ligand atom positions in coordination compounds and by the core and by the peripheral atoms in polyhedral boranes and metal clusters. Examples of polytopal isomers are the tetrahedron and square plane in the four-atom family, and the square antiprism and trigonal dodecahedron in the eight-atom family. The term polytopal isomer will carry no connotation of lifetime as in structural and configurational’ isomers. A polyhedral rearrangement is defined as a traverse of two or more polytopal forms, i.e., deformation of a specific polytopal form, by thermal or photochemical excitation, to the point that the ligand atoms define a new polytope. The traverse presents the possibility of permutation of ligand atom position and accordingly stereoisomerization within either polytopal form. An example is the racemization of an optically active nickel(11) tetrahedral complex through deformation to a square-planar form. Many molecular species undergo polyhedral rearrangements, and there is extensive research on stereochemically nonrigid or fluxional molecules. Recent review articles or books cover most of the major earlier related p a p e r s : (a) E . L. Muetterties, Inorg. Chem., 6 , 635 (1967); (b) J . Am. Chem. Soc., 90, 5097 (1968). (2) K. Mislow, “Introduction to Stereochemistry,” W. A. Benjamin, Inc., New York, N. Y., 1965, p 51. (3) Definition of idealized polygons and polyhedra by atom positions is applicable to the majority of molecular species. Although this terminology is imprecise in those cases where the atoms are nonidentical, the ligand atom positions do generally conform closely to idealized polyhedra. This definition is employed to maximize symmetry and simplify discussions. (1) CJ
Journal of the American Chemical Society 1 91:7 1 March 26, 1969
In the following section, reference will be made to the “general case.” This is a hypothetical stereochemical system of the type ML, or M, wherein ligand-atom or core-atom vertex sites are nuclearly distinguishable by labeling. This is done simply to engender high symmetry in the topological approach.
Definitions
A specific x-atom problem is described as a stereochemical family, F, composed of all stereoisomers of all polytopal isomers GA,G,, Gc, . . . To simplify the analysis, the family will be factored out into classes Y whose members are the stereoisomers of two polytopal sets, GA-GB, GA-Gc, etc. An operation, 0,a specific change in shape parameters, is postulated to alter the spatial arrangement in ML, or M, from one idealized polytope (GA) to a second (GB). Two sequential operations performed on a stereoisomer of the GA (or GB)set will generate a new stereoisomer of this set. The 0.0’ operations will be referred to as steps, R , or processes. The step order of a specific stereoisomerization is simply the number of steps involved. The product 0 x 0-’ is simply the identity process. Connectivity, 6, of polytopal forms, referenced to a postulated process, is the number of new stereoisomers of GA (or GB) generated from a stereoisomer of G, (or GB) in one step. Connectivities of individual stereoisomers within the G, (or GB) set are always identical in a general ML, or M, class but may vary in others. A stereochemical class, 9, may be open or closed in a graphical or topological sense. Closure obtains if and only if the operations permit traverse of all stereoisomers in the two polytopal classes. Graph theory requires for a closed system the condition that the product of the isomer count and the connectivity of one polytopal set equals that product for the other set, i.e., ZA8* = ZB8B. However, one polytopal set may be closed for the given operation even though the products are not equal (vide infra). A connectivity of one for either set indicates an
.
(4) E. L. Muetterties and R. A. Schunn, Quarr. Rev. (London), 20, 245 (1965). ( 5 ) F. W. Westheimer, Accounts Chem. Res., 1, 70 (1968). (6) E. L. Muetterties and C . M. Wright, Quart. Rev. (London), 21, 109 (1967). (7) W. N. Lipscomb, Science, 153,373 (1966);‘ (8) E. L. Muetterties and W. H. Knoth, Polyhedral Boranes,” Marcel Dekker, Inc., New York, N. Y., 1968, Chapter 111. (9) G. Schroeder, Angew. Chem. Intern. Ed. Engl., 4, 752 (1965). (10) G. Schroeder and J. F. M. Oth, ibid., 6, 414 (1967). (11) F. A. Cotton, Chem. &if., 4, 345 (1968).
1637 open system unless the Z.6 product of the other set is equal to I for the former. If isomer connectivities vary within a set, then the sum of the individual products is taken. The isomer count, Z, in the general case for a specific polytope is simply N !/hr,where N is the number of polytopal vertices and h, is the order of the rotational subgroup of the point group of the polytope. (I am indebted to Professor F. A. Cotton for this explicit expression.)
Topological Analysis Given the above definitions and the postulation of a stereoisomerization problem, the stereochemical class can be analyzed by graph theory. This analysis is quite straightforward if the number of isomers is not too large (at least up to ML, where I = 20-30), but only computer analysis is a realistic approach for the more complicated systems. Several specific elements in the topological analysis are of particular interest for application to mechanistic problems. They are cycles $9, subcycles C, and chains c. These elements are defined as follows with their order equal to the number of steps. Cycle V is a circular path that traverses an isomer a and its enantiomer a. No isomer is traversed more than once in the circuit of a cycle. The cycle of minimal order is designated gE. Subcycle C is a circular path that does not traverse enantiomeric isomers; Le., it passes through a, b, c, etc., but not a, b, C, etc. A trivial subcycle of order two is generated by the steps R.R-'.12 Chain c is any noncircular path. There may be unique chains ; one of specific stereochemical importance is the chain of cE, of minimal order for the traverse from a to a. Important characteristics of these elements are their degeneracy, e.g., number of or cE's per isomer and C's of given order per isomer, and the chirality distinctions among elements. A stereochemical class may be described by a process table or matrix for either of the two polytopal sets. The matrix explicitly characterizes the minimum number of steps from any isomer to a new isomer (Tables I and 11). The matrix, unfortunately, becomes unwieldy if isomer count is large. Some subcycles are not immediately apparent by inspection of the table but they are derivable. In any case, the matrix is readily generated and may be the only recourse if the class cannot be described in a projection of a three-dimensional, noninterpenetrating figure (vide infiu). Topological illustrations are desirable in two- and three-dimensional representations because all pathways and the degeneracies of $9, C , and c elements are immediately apparent. Furthermore, such illustration can be used for energy relationships and, in principle, if energy data are available, for construction of potential energy diagrams. Unfortunately, there are severe limitations. Regular three-dimensional polyhedra are few (five platonic solids and four stellated icosahedra and (12) There always appears to be a subcycle of order one greater than %'~/2. This is simply based on analyses of a number of stereochemical classes.
pentagonal dodecahedra). Even if the isomer count can be accommodated in such regular polyhedra, the subcycle requirements are not always satisfied. Some representations require higher dimensionality and cannot be properly reduced for illustration.' For stereochemical classes of high isomer count (in MX,-D,,, Z = 504), process matrices, networks, or graphs would be too large, and explicit topological illustration is not feasible. Elements (cycles, etc.) can, however, be obtained by computer analysis and so define the individual systems. Following is an illustration of this approach and of problems encountered in generating proper representations.
Illustrative Examples Cyclic Groups. If the connectivity of GA and GB is two in a closed class, the operations comprise a cyclic group. The class may be represented simply by a p-gon (C = Y ) , where p is equal to the number of isomers. Many stereochemically labile organometallics meet these conditions. For example, it has been established through nmr studies'' that a cyclopentadienyl group singly bonded to a metal is nonrigid; the carbon atoms attached to the metal atom are rapidly permuted. Permutation might occur through a 1,2 or 1,3 shift. The cyclic group associated with these operations (rotations) is C,; the group elements are r" (n = 1-5), where r is a rotation of 72". Representations of these two possibilities would then be regular pentagons with vertex numbering (labels for carbon atoms bound to the metal) of 1-5 and 1, 3, 5, 2,4, respectively. Note that for a 1,3 shift the alternative representation (which is not a planar graph) shown in Figure 1 cannot be converted to an energy surface because of the crossover of reaction paths. In the planar graphs, vertices represent GA isomers and edge midpoints the GB isomers; the connecting segments are the operations. The graphs can be converted, in principle, to potential energy diagrams by the proper deformation of line segments. In cyclopentadienylmetalates, permutation of carbon vertices via a "sandwich"-type intermediate does not yield a cyclic group since 6 for the G B intermediate or transition state is 5. Here the representation is a tree with the G, transition state connected to five points representing GA isomers. Closed Stereochemical Classes. General Cases. Threecoordinate complexes may have pyramidal or coplanar form. These two polytopal forms are related by a simple MX bending mode. Topologically this class is trivial. The preferred representation is spheroid, either oblate or prolate, depending upon which polytopal form is lower in energy. The d and 1 pyramidal isomers are placed at the antipodal positions and the coplanar form is at the equator.I3 The two idealized polytopal isomers for four-atom systems are the tetrahedron ( I = 4!/12 = 2) and the square plane ( I = 4!/8 = 3). Conversion of tetrahedron into a planar form may be visualized as a twisting of two mutually orthogonal planes of the tetrahedron, each (13) For a recent discussion of inversion in pyramidal molecules, cy. G. W. Koeppl, D. S. Sagatys, G. S. Krishnamurthy. and S. I. Miller, J. Am. Chem. SOC.,89, 3396 (1967).
Muetterties 1 Topological Representation of Stereoisomerism
1638
FOUR ATOM -T,j-D4h-Djh
TORUS
ISOMERS- 3 CONNECTIVE
Td-2
Dqh- 3 ISOMERS
-2
CONNECTIVE
*Doh
Figure 1. A nonplanar graph for the 1,3 shift process in cyclopentadienylmetalates. Reaction paths are indicated by dotted lines.
FOUR ATOM
-Td-Dqh -D3h
TRIGONAL
BIPYRAMID
Td - TWO ISOMERS - 3 CONNECTIVE D4h- THREE ISOMERS- 2 CONNECTIVE
Figure 3. “Toral” representation of the Td-Dlhclass. The three figures at the bottom are cross sections to illustrate relative energy relationships as in Figure 2.
- HEXAGONAL
FOUR ATOM -Td-Dzh-DGh Td Td
ISOMERS -6 CONNECTIVE
D2h- SIX ISOMERS- 2 CONNECTIVE
Figure 2. Topological representation of the T d - D 4 h class. Polyhedra at the bottom illustrate three possible representations in an energetic sense with the one on the left and the one on the right illustrating Td and D4hground states, respectively.
through an angle of 45”. Operationally, the tetrahedron has a connectivity of three and the square plane two. The 1 . 6 products are identical. A topological representation is a trigonal bipyramid of D,, symmetry (Figure 2). Edges connecting equatorial vertices are disallowed reaction paths, There are no subcycles in this class. The chain cE for the tetrahedral form is first order, one step for d to 1 conversion. Relative placement of the Td and D,, energy levels is achieved as shown at the bottom of Figure 2 by compression or elongation of the bipyramid along the threefold axis. The elongated form is representative of heavy-atom d8 complexes, the intermediate form of some cobalt and nickel(I1) complexes, and the compressed form of most chemical systems. An alternative topological representation, derived from a torus, is depicted in Figure 3. The related T,-D,, class is also closed, and a hexagonal-bipyramidal representation is given in Figure 4. Dihedral Groups. Two idealized polytopal isomers in the six-atom family are the octahedron and the trig: onal prism which may be interconverted by counter face rotation on a threefold axis. In the general chelate class M(chel),, there are two possible operations: one a twist on a real threefold axis and one on an imaginary
- TWO
BIPYRAMID
a
d
C
D2h ISOMER P L A N E
d‘
% Td
Td
Figure 4. Topological representation of the Td-DZhclass.
threefold axis (Figure 5). These operations form a dihedral group. The two kinds of operations yield two different types of trigonal prismatic isomers designated Bailar and Ray and Dutt as shown in Figure 5. A rrans span of a square face is not allowed. For this class, the I.6 products are 16 x 4 and 32 x 2 for the octahedron and trigonal prism, respectively. An explicit topological representation is the cell-first reduction of the y4 (tesseract) polytope (Figure 6 ) with the octahedral isomers at vertices, the Ray and Dutt type intermediates at edge midpoints of the two interconnected cubes, and the Bailar intermediates at the midpoints of the lines con-
Journal of the American Chemical Society / 91:7 / March 26, I969
1639 PROCESS No.
-@
:-.
e'
"S
a
-3n I , a
I,
a a
0
112 iRC3
Figure 5 . Operations for the M ( ~ h e 1 )class. ~ At the top and bottom right are the Bailar and the Ray and Dutt intermediates, respectively.'
a
4
e'
e
e+
c
e+
D3h
e+ D3h
a'
a
a'
a'
P
Figure 7. Possible processes for isomerizations in the five-atom class. 'I
Figure 6 . Topological representation of the dihedral stereoisomeric group M(chel),.
necting the two cubes. Elimination of the RC, process leaves two isolated subsets which are cubes. In these groups racemization is allowed. Elimination of the iRC, process leaves eight isolated chains of order one. From Figure 6 it can be seen that cycles WE,are of d4 =+I, -+ order six and of two types, e.g., d, 5 I , d5 5 I4kd, and d, =+ I , %- d , =+I, f+ d4=+ la -+ dl. The d and 1 designations are arbitrary in that they do not specifically refer to dextro and levo forms. There are also cycles of order eight as in I , -+ d2 --+ I, -+d3 -+ l2 -+ d, -+ l5 -+ d , -+ I, and of order ten as in I, --+ d , -+ 1, -+d , -+ l4 -+ d, -+ I, -+ d6 -+ 1, -+ d4 -+ 1, (note the pervasive alternation in chirality for all operations of these stereoisomeric elements). Subcycles are all tetragons but there are two types. One element of 12-fold degeneracy is represented by cube faces (all steps of + - + - sequence, and all involve CZvtrigonal prismatic intermediates; Le., all steps are iRC, as in I, -+ d , -+ I, --+ d4 -+ I,). The second element (e.g., l5 % d6 =+1, b d4 > I,) has 12-fold degeneracy and has the same chirality alternations as the other subcycle, but trigonal prismatic pathways alternate between D,, and
+
+
CZvforms. Chains, cE, are of order three, Le., traverse four isomers and comprise three steps (+ - + or - -) -and are only six degenerate with respect to each enantiomeric set.14
+
Representation Difficulties. Stereochemical Matrices The trigonal bipyramid and the square pyramid are the prevailing polytopal isomers in the five-atom family. A mode for interconversion of these forms, first proposed by Berry,'5 is process 1 of Figure 7. The 1 . 6 products are 20 x 3 and 30 x 2 for the D,, and C,, forms, respectively; the stereochemical class is closed. Class elements are the cycle W E of order ten, the subcycle of order six with threefold degeneracy, and the chain cE of order five with 12-fold degeneracy. The regular pentagonal dodecahedron with D,, isomers at the 20 vertices and C4visomers at the 30 edge midpoints is not a solution. One obvious deficiency is the absence of hexagons for the subcycle requirement. This can be resolved with a double group representation and quasiRC5 operations,lb but this does not yield a conceptually simple model. No three-dimensional representation can (14) Compare with the cE of order three and degeneracy of 24 in MLslb (the reduction in degeneracy is due to the chelate constraints; only one direction of "twist" in the iRCJ operations is allowed); this chelate constraint also accounts for the loss of the subcycle or order 3. (15) S. Berry, J. Chem. Phys., 32, 923 (1960).
Muetterties
Topological Representation of Stereoisomerism
1640
R~
12
23
34
45
13
14
15
24
25
35
i2
23
34
43
i3
12 23 34 45 13 14 15 24 25 3
E 2 4 4 3 2 3 3 2
2 E 2 4 3 4 1 3 2
4 2 E 2 3 2 1 3 4
4 4 2 E 1 2 3 3 2
3 3 3 1 E 3 2 4 1
2 4 2 2 3 E 3 3 4
3 1 1 3 2 3 E 4 3
3 3 3 3 4 3 4 E 3
2 2 4 2 1 4 3 3 E
1 3 3 3 2 1 2 4 3
5 3 1 1 2 3 2 2 3
3 5 3 1 2 1 4 2 3
1 3 5 3 2 3 4 2 1
1 1 3 5 4 3 2 2 3
2 2 2 4 5 2 3 1 4 3
23 34
45
i3
_i154_ 24 25 35
i4
3 1 3 3 2 5 2 2 1 5 1 3 3 3 2 1 2 4 3 E 4 2 2 2 4 5 3 1 1 2 3 2 2 3 4 ~ 2 4 4 3 3 5 3 1 2 1 4 2 3 2 2 E 2 4 3 4 1 3 5 3 2 3 4 2 1 2 4 2 E 2 3 2 1 1 3 5 4 3 2 2 3 2 4 4 2 E l 2 2 2 2 4 5 2 3 1 4 3 3 3 3 1 ~ 3 2 1 4 2 4 2 2 3 € 3 1 3 3 2 5 2 3 2 4 4 2 3 2 5 1 2 3 3 1 1 3 2 2 2 2 2 1 2 1 5 2 1 3 3 3 3 4 3 3 3 1 3 4 1 2 2 5 2 2 2 4 2 1 4 4 2 2 2 3 4 3 1 2 5 1 3 3 3 2 1
be derived to satisfy all of the requirements without recourse to an interpenetrating figure. In this instance, a stereochemical matrix is preferred for simplicity. The matrix for the trigonal-bipyramid group is given in Table I with the entries representing the number of steps (R)to go from an isomer in the vertical column to one in the horizontal column. Specific isomers are designated by the axial substituents, e.g., 12 and its enantiomer E. All cycles, subcycles, and chains are readily obtained from the matrix, This type of presentation is recommended also for the general ML, class; above six the order is too large for matrices to be reasonably employed. A distinction in mechanisms is immediately apparent by comparing the matrix for the Berry process with that for process 4 of Figure 7. This latter process has Z.6 products of 20 x 6 and 120 x 2 for D,, and C,, respectively; the class is open. The class consists of two disjoint and isomorphic (enantiomeric) groups. The stereochemical matrix for one of the groups is given in Table 11. Note that racemization cannot occur by this mechanism, and that there are many subcycles of order three, four, and five. Processes 4 and 6 are identical. Process 2 yields an open class but is closed f o r the trigonal-bipyramid set (Table 111). A detailed discussion of isomerization mechanisms in five-coordination is deferred to paper I1 of this series.
Stereoisomerization. Differentiation of Mechanisms Eight-Atom Family. Polytopal isomerism is quite complicated for the eight-atom family in that there is a large number of possible isomers. Of the depicted isomers in Figure 8, only the D,, isomer has not been reported for a molecular species. In coordination compounds and polyhedral boranes, the square antiprism (D4,) and trigonal dodecahedron (D,,) appear to be nearly equivalent in energy., Possible operations for interconversions of some of these forms are illustrated in Figures 9-12. Isomer counts for Oh, D,,, DZd,and CZv are 1680, 5040, 10,080, and 20,160, respectively.
Journal of the American Chemical Society / 91:7 1 March 26, 1969
i3
24
25
35
2 4 4 2 3 2 5 1 2 3 2 1 1 3
2 2 2 2 1 2 1 5 2 1 3 3 3 3
3 3 1 3 4 1 2 2 5 2 3 2 4 2 4 4 3 3 E 3
4 2 2 2 3 4 3 1 2 5 2 3 3 3 1 1 2 4 3 E
2 3 E 4 3 2
3 4 E 3 4
1
2
Table 11. ML6-Process 4
12 E 2 5 1 iJ I 24 1 13 1 23 1
1 1 2 1
E
2
2 1 2
E
45 34
1 2
1 2
2 2
1
1 E
1
2 1
1 2 E 1
1 1 2 1 1 E
1 2 1 1 1 2
2 1 1 2 1 1
2 1 I 1 2 2
2 2 2 1 1 1
1 1
2 1
2 1
1 1
1 1
E 1
1 E
1
1 1
2
Based on the illustrated operations, the connectivities are 0h(-fD4d) = D4d(+0h) oh(-fD2d) = D4d(-fD2d) = D4d(-fCZv) = DZd('Oh) = D2d(-fD4d) = C2~(-fD4d)= The connectivities of one for D,, in D,, P Oh interconversion and for C,, in D,, P Czyshow the openness of these classes; they may be treated as isolated subsets represented by 1680 octahedra (D,, isomers at vertices and an 0, at the center) and by 5040 tetrahedra, respectively. Racemization cannot occur in either class. The 0, P D,, and D4, P Oh classes are closed. In the former, the racemization chain cE is of order six with one ( +)'/,RC, and eleven ( -)'j2RC4, or one ( -)1/2RC4 and eleven (+)'j2RC, operations. One of the subcycles is seventh order. For the D,, F? D,, system c E is 13, and there are subcycles of order 2n with n = 3-7.16 Twelve-Atom Family. For the 12-atom family the prevailing ground-state form is the icosahedron (1,) in coordination compounds and in polyhedral boranes. An operation6'* on the icosahedron that ultimately leads to isomerization is a stretching of three orthogonal sets (16) I am indebted to Dr. J. P. Jesson for writing the computer program for this permutation problem.
1641 Table 111. ML,-Process
12 23 34 45 13 14 15 24 25 35
iz 25
34 45
24 25 33
E 2 2 2 1 2 1 1 2 3 3 1 3 3 2 1 2 2 1 2
2 E 2 2 1 2 3 1 2 1 1 3 1 3 2 3 2 2 1 2
2
2 2 E 2 1 2 3 1 2 1 3 1 3 1 2 1 2 2 3 2
2 2 2 E 3 2 1 1 2 1 3 3 1 3 2 1 2 2 1 2
1 1 1 3 E 1 2 2 3 2 2 2 2 2 3 2 1 3 2 1
2 2 2 2 1 E 1 1 2 3 1 2 1 1 2 3 2 3 3 2
1 3 3 1 2 1 E 2 1 2 2 2 2 2 1 2 3 3 2 1
1 1 1 1 2 1 2 E 1 2 2 2 2 2 3 2 3 3 2 3
2 2 2 2 3 2 1 1 E 1 1 1 3 1 2 3 2 2 3 2
3 1 1 1 2 3 2 2 1 E 2 2 2 2 1 2 1 3 2 3
3 1 3 3 2 1 2 2 1 2 E 2 2 2 1 2 1 1 2 3
1 3 1 3 2 2 2 2 1 2 2 E 2 2 1 2 3 1 2 1
3 1 3 1 2 1 2 2 3 2 2 2 E 2 1 2 3 1 2 1
3 3 1 3 2 1 2 2 1 2 2 2 2 E 3 2 1 1 2 1
2 2 2 2 3 2 1 3 2 1 1 1 1 3 E 1 2 2 3 2
1 3 1 1 2 3 2 2 3 2 2 2 2 2 1 E 1 1 2 3
2 2 2 2 1 2 3 3 2 1 1 3 3 1 2 1 E 2 1 2
2 2 2 2 3 3 3 3 2 3 1 1 1 1 2 1 2 E 1 2
1 1 3 1 2 3 2 2 3 2 2 2 2 2 3 2 1 1 E 1
2 2 2 2 1 2 1 3 2 3 3 1 1 1 2 3 2 2 l E
Oh
Figure 10. Distortion of the cube to generate the trigonal dodecahedron.
w ......
Figure 1 1 . Operation for square antiprism to Cz,-bicapped trigonal prism traverse.
Figure 8. Idealized polytopes in the ML8 or Ma family.
Figure 9. An operation for traverse of D4,, and Oh polytopes in the eight-atom class.
Figure 12. Operation for Cz,-bicapped trigonal prism to trigonal dodecahedron traverse.
of two coplanar edges. An intermediate state is the patohedron6>*which has isosceles triangular faces and Th symmetry; see Figdre 13 for the stretching mode. The icosahedron cannot be isomerized via the patohedron because the connectivity of the latter is one. Hence the
Ih-Th class is open and consists of disjoint groups, namely 7,983,360 trees with an 1, isomer connected to five Th isomers. If the stretching mode outlined in Figure 13 is carried to the point that square faces (six) are generated, cube octahedral geometry (0,) results
Muetterties Topological Representation of Stereoisomerism
1642
Figure 13. Distortion mode for I,, + Th.
d4
d5
Figure 16. Stereoisomers in three polytopal isomer forms of M(chel),.
Different Isomer Connectivities Figure 14. Isomerization of icosahedral ML,, or M12 species via the cube octahedron. Antipodal relations are maintained throughout the process.
In eight-coordinate chelates of the type M(chel),, there are two predominating polytopal isomers, the square antiprism (“D4d”)and the trigonal dodecahedron (“DZd”). Intermediate between these two polytopal isomers is the bicapped trigonal prismatic (“Czv”)form (see Figures 11 and 12). The antiprism-dodecahedron and bicapped trigonal prism-dodecahedron stereochemical classes are closed ; individual stereoisomer connectivities vary. The stereoisomers are depicted in Figure 16. First-order operational relationships for these two classes are as fo!lows. “D4d(’
Figure 15. Face rotation mechanism for isomerization in icosahedral MLlz or M I Z species.
(Figure 14). The isomer counts are: Z(1,) = 12!/60 = 7,983,360 and Z(0,) = 12!/24 = 19,958,400. Connectivities are five and two for Ih and oh, respectively. The stereoisomeric class is not closed since Z.6 products are not equivalent. This openness is also visually evident from inspection of the process depicted in Figure 14. Antipodal relationships of vertices (1-12, 2-9, etc.) must be maintained in both polytopal forms. Interconversion of enantiomers may occur by this process; the minimum number of steps for this process is seven. An operation that yields a closed class for the icosahedron is single-face rotation (Figure 15). The intermediate polytopal form has C, symmetry, an isomer count of 159,667,220, and a connectivity of two. The connectivity of the icosahedron with respect to this operation is simply twice the number of faces in this polyhedron since the faces can be rotated in either direction. Thus the equality I*6(Ih)= 7,983,360 x 40 1 = 319,334,400 I * 6 ( C 3 )= 159,667,200 x 2f is obtained. Journal of the American Chemical Society 1 91:7 March 26, I969
“Dzd”
di
(11)
diJ2
d2
(12)
1 1 4 3 , ml
(lid21
MAmd d 3 , d d m (k,l4,m2)
d3 ( 1 3 )
‘‘D4d” xZi6i = (2
X
“Dzd” cZi6i = (1
X
+ (2 X 3) = 16 1) $- (1 X 2) + (2 X 2) + (2 X 1) + (2 x 2) + (2 x 1) = 16
2)
+ (2 X
3)
Two specific first-order operations are illustrated in Figure 17.
“Dzd”
+ (2 x
+ (2 x 2) + (2 x 2) + (2 x 2) = 20 CZi6i = (1 X 2) + (1 X 2) + (2 X 2) + (2 x 1) + (2 x 3) + (2 x 2) = 20
“CZV”CZJi = (2 x 2)
2)
In contrast, to the above two classes, neither the D4,-Oh nor the Dzd-O, classes in M(chel), are closed, and within the subsets isomer connectivities vary.
1643
D2d
D4
D2
02
Figure 17. Illustration of dodecahedral to square antiprism operations for specific stereoisomers: (top) Dzd(“ml”)+ D4(“di”); (bottom) Dzd(“dl”)+ D2(“d2”).
Figure 19. Trigonal prismatic stereoisomers in M A ~ B ~ C Z .
A
A
A
Figure 18. Octahedral stereoisomers in M A ~ B z C ~ .
Another stereochemical system with isomers of varying connectivity is the six-coordinate MA,B,C, class with Octahedral and trigona1 prismatic polytopal forms* The Processes are RC3 trigonal twists on c3 and ic3 axes. The 6 octahedral and 18 trigonal prismatic isomers are shown in Figures 18 and 19. First-order operational relationships are “Oh” di mi
mz
m3
m4
“Oh” Z.6 =
duality relationship, the octahedron may be placed symmetrically within a pentagonal dodecahedron (Figure 20). The octahedral vertices represent the 6 octahedral isomers, and 18 of the dodecahedral vertices are used for the trigonal prismatic isomers.”
i‘Djh’’ ml, m 2 ,m3,m4, dl,d2,d3,d4 d4,14, ds, 1 5 , d691.5
d3, 13, de,16, d7,17 dz, 12 di, 11, dsg 1s ds, I6
Zi6i = (2 x 8)
+ (1
-t
‘‘D3h”1.6 = CiIiSi = 4(1 x 2)
x 6)
2,
+
+ 7(2 x
+ (1
Figure 20. Topological representation of the MA2B2C2 class. Octahedral stereoisomers are at the vertices of the enclosed octahedron and trigonal prismatic stereoisomers at 18 of the 20 pentagonal dodecahedral vertices. First-order isomer relationships are shown by line connectors between the octahedron and the pentagonal dodecahedron. The figure is an interpenetrating one.
x 6)
(l 6, = 36 2) = 36
The 1 . 6 products are equivalent and the class is closed. representation Topologicallyy a is achieved with a compound polytope. Because of the
Acknowledgment. I am indebted to Mr. Ames Schuck for the drawings and to Professor John Wheeler and Dr. J. P. Jesson for invaluable discussions. (17) NOTEADDED IN PROOF. Closure of a stereochemical class requires the connected graph condition of I . 6 product equality and a rearrangement that is an odd permutation if the group of all permutations representing the isomers is even-odd: E. L. Muetterties and A. T. Storr, to be published.
Muetterties / Topological Representation of Stereoisomerism
